Methods in Health Services Research

.title[
# Methods in Health Services Research
]
.subtitle[
## Lecture 1: Introduction
]
.author[
### Jacob Wallace
]
.date[
### Yale School of Public Health | <a href="https://yale.instructure.com/courses/93126">HPM 583</a>
]

---

# Table of contents

1. [Welcome and motivation](#prologue)

2. [Warmup exercise](#equation)

3. [Course details](#controls)

4. [Introduction to causal inference](#inference)

---
class: inverse, center, middle
name: prologue

# Welcome and motivation

---

# The "big data" revolution has come, so what?

- There is now more data than ever before
  - Over the last two years alone, what % of the data in the world was created?
  - How much of the world's stored data is generated in the healthcare industry?

---

# The "big data" revolution has come, so what?

- Facebook, Amazon, Apple, Netflix, and Google (FAANG) have large **data science** teams that use data data to learn about the world and drive business decisions

.center[
<figure>
  <img src="images/faang.jpeg" alt="Camden coalition" title="Camden coalition" width="40%">
</figure>
]

<br>

- Nonprofits, governments, and healthcare organizations are increasingly focused on how to harness healthcare data to make better decisions, policies, and products

- Well that all sounds great, right?

---

# It's true that we have a lot of data in healthcare

### Problem Set 2
  - Due February 21 at 3pm (~a week from today)
  - Reminder: You can work together, ask for help, etc.

### Midterm 1
  - Will be in-class on March 7th
  - Covers material from the readings, lecture, and lab
  - We will schedule a review session prior to the midterm

---

# Estimate = Estimand + Bias + Noise

---

# Recap

- Ordinary least squares is the practice of fitting a line to data so as to explain the relationship more generally
- It picks the straight line that minimizes the *sum of squared residuals*

---

# Recap

---

# Ordinary least squares

- The line of best fit has:
  - **intercept** of `$\beta_{0}$`, our prediction of `$\hat{Y}$` when `$X=0$`; and
  - **slope** of `$\beta_{1}$`, how much higher we predict `$Y$` will be when `$X$` is one unit higher

- The *residual* is the difference between our prediction `$\hat{Y}=\hat{\beta_{0}}+\hat{\beta_{1}}X$` and the actual number `$Y$`.

`$$\begin{equation}
Y=\beta_{0}+\beta_{1}X+\varepsilon
\end{equation}$$`
`$$\begin{equation}
\hat{Y}=\hat{\beta_{0}}+\hat{\beta_{1}}X
\end{equation}$$`

---

# Noise and bias

- We have an idea that there's this *true* relationship `$Y=\beta_{0}+\beta_{1}X+\varepsilon$`

- We can call that relationship the *data generating process*: The underlying equation that explains where our observations of `$Y$` come from

- What we are trying to do by estimating OLS is to get estimates `$\hat{\beta_{0}}$` and `$\hat{\beta_{1}}$` that are as close as possible to `$\beta_{0}$` and `$\beta_{1}$`

- But, we must be on the lookout for two sources of error in our estimates...

![](gifs/lookout.gif#center)

---

# Estimate = Estimand + Bias + Noise

- There are two main sources of error in our estimates
  - One is *bias* (endogeneity) which is systematic
  - The other is *sampling variation* which is random

Consider the estimate `$\hat{\beta_{1}}$` of the estimand `$\beta_{1}$`:

`$$\begin{equation}
\hat{\beta_{1}}=\beta_{1}+\text{Bias}+\text{Noise}
\end{equation}$$`

Today we begin by discussing how we can use multiple linear regression to try and control for **bias** and then discuss *sampling variation* as a source of **noise**

---

# Bias

---

# Bias

- Last week we used DAGS to help identify confounding and discussed the **backdoor criterion** for isolating a causal effect

- This week we're going to talk about how to use multiple linear regression to control for those pesky confounders

---

# Endogeneity recap

- Suppose we believe that our true model looks like this:

`$$Y = \beta_{0}+\beta_{1}X+\varepsilon$$`

- where `$\varepsilon$` is everything that determines `$Y$` other than `$X$`

- If `$X$` is related to some of those things, we have **endogeneity** (or bias)

- Estimating the above model by OLS, it will mistake the effect of those *other* things for the effect of `$X$`, and our estimate of `$\hat{\beta_{1}}$` won't represent the true `$\beta_{1}$` because of this bias.

- **Concept check:** If `$Y$` was "height in feet" and `$X$` was "height inches", what would be in the error term?

---

# Shorts and ice cream

Consider the following model:

`$$IceCreamEating = \beta_{0}+\beta_{1}ShortsWearing+\varepsilon$$`

- Surely `$ShortsWearing$` isn't the *only* thing that determines your `$IceCreamEating$`, in fact the true `$\beta_{1}$` is probably 0.

- Everything else is in the error term!

- `$Temperature$`, `$Income$`, `$Age$`, and so on and so on...

---

# The Error Term

`$$IceCreamEating = \beta_{0}+\beta_{1}ShortsWearing+\varepsilon$$`

- Isn't it really bad to leave out a bunch of important stuff? Depends on the goal...

- If you want to *predict `$Y$` as accurately as possible* then we're probably going to do a bad job of it

- But if our goal is to estimate the causal relationship between `$X$` and `$Y$` then it can be okay to leave some stuff out

- The latter goal - understanding **causal relationships** - is what econometrics is more concerned with

- The former goal - **predicting accurately** - is more the domain of data scientists

---

# Error Term Assumptions

- The most important assumption about the error term is that it is *unrelated to `$X$`*

- If `$X$` and `$\varepsilon$` are correlated, `$\hat{\beta}_1$` will be *biased* - its distribution no longer has the true `$\beta_1$` as its mean

- In these cases we can say "we have **endogeneity**" or "we have **omitted variable bias**" or "we have **confounding**" or "we have **selection bias**"

- No amount of additional sample size will fix that problem!

---

# But we have a way of dealing with it

![](gifs/linear_regression.gif#center)

Source: [Twitter](https://mobile.twitter.com/theotheredmund/status/1349453230762196992)

---

# Epilogue

---

# Where are we going

- Next class: More on how to use linear regression to estimate causal effects
- Problem set 2 is due February 21
- **Midterm 1** will be in-class on March 7th

<br>

![](gifs/thatsall.gif#center)